Nuprl Lemma : copath-extend_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. ∀[p:copath(a.B[a];w)]. ∀[t:coW-dom(a.B[a];copath-at(w;p))].
(copath-extend(p;t) ∈ copath(a.B[a];w))
Proof
Definitions occuring in Statement :
copath-extend: copath-extend(q;t),
copath-at: copath-at(w;p),
copath: copath(a.B[a];w),
coW-dom: coW-dom(a.B[a];w),
coW: coW(A;a.B[a]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
top: Top,
subtype_rel: A ⊆r B,
subtract: n - m,
squash: ↓T,
sq_stable: SqStable(P),
uimplies: b supposing a,
uiff: uiff(P;Q),
prop: ℙ,
false: False,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
nat: ℕ,
copath-at: copath-at(w;p),
copath: copath(a.B[a];w),
copath-extend: copath-extend(q;t),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
coW_wf,
copath_wf,
copath-at_wf,
coW-dom_wf,
coPath_wf,
coPath-extend_wf,
le_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
sq_stable__le,
not-le-2,
false_wf,
decidable__le
Rules used in proof :
universeEquality,
functionEquality,
cumulativity,
instantiate,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
lambdaEquality,
applyEquality,
imageElimination,
baseClosed,
imageMemberEquality,
isectElimination,
independent_isectElimination,
independent_functionElimination,
voidElimination,
lambdaFormation,
independent_pairFormation,
unionElimination,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
natural_numberEquality,
hypothesis,
because_Cache,
rename,
setElimination,
addEquality,
dependent_set_memberEquality,
dependent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])]. \mforall{}[p:copath(a.B[a];w)].
\mforall{}[t:coW-dom(a.B[a];copath-at(w;p))].
(copath-extend(p;t) \mmember{} copath(a.B[a];w))
Date html generated:
2018_07_25-PM-01_39_12
Last ObjectModification:
2018_07_18-PM-07_49_38
Theory : co-recursion
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